I am pronouncing the word Pǔtōnghuà 普通话/普通話 as [pʰ᷆u˨˩·tʰʊ́ŋ˧˥·xû̯ä˥˩] with the first syllable being pronounced with a 低降调 low falling tone (調值 21). However, I was told that I should pronounce the first syllable with a 低降升调 low falling-rising tone (調值 214) like this [pʰ᷉u˨˩˦·tʰʊ́ŋ˧˥·xû̯ä˥˩].

I do believe I am using the correct tone, but I am learning Mandarin, whereas the person telling me I am wrong is a native Chinese speaker.

I do understand the pīnyīn spelling includes a caron on the first syllable and the caron represents the 3rd tone in Mandarin Chinese. However, since the first syllable is followed by a syllable with a 阴平声 high tone (調值 55), then wouldn't the first syllable be pronounced with a 低降调 low falling tone (調值 21)?

In short, you're correct in pronouncing the first syllable in low falling tone. However the person who told you it should be rising-falling is not completely wrong either.

The third tone (第三声 or 上声, the one represented by the caron) strictly speaking really a falling-rising tone (214 for standard Mandarin, just as you said). However it is only pronounced in that way if it's not followed by anything else. If a syllable in third tone is followed by a syllable in any tone but the third (1st, 2nd, 4th or light tone/toneless syllable), it will be pronounced in what some people call the "half-third tone." This concretely means that the rising part is left out, producing your low falling tone. If a syllable in third tone is followed by another syllable in third tone, the first one will be pronounced in the same way as the 2nd tone (第二声 or 阳平声), i.e. as a high rising tone (35). Therefore, for example, 你好 (nǐ hǎo) is pronounced as "ní hǎo". Note however that this so called "tone sandhi" ("sandhi" is a Sanskrit term meaning "joining") is not usually reflected in Pīnyīn notation, readers are expected to know when to change the tone.